The correlations achieved by separated measurements on entangled particles measurably exceed what is classically achievable. That's what's meant by "Bell inequality violations".

We can't transmit information instantaneously, but there are nevertheless certain distributed tasks we can do better at when we have a source of entangled states.

Better than just using a pseudorandom number generator with the same seed?

Yes. Down on the Wikipedia page there is a fairly basic game that you can convince yourself is impossible to win with 100% probability without communication: https://en.wikipedia.org/wiki/Quantum_pseudo-telepathy#Green...

To paraphrase, Alice, Bob and Carol play a game where they can't communicate (after deciding on a strategy) and the referee shows each of them a bit. After being shown the bit they must reply with 0 or 1. The four possible combinations the referee chooses uniformly from is:

    (Alice: 0, Bob: 0, Carol: 0)
    (Alice: 1, Bob: 0, Carol: 1)
    (Alice: 1, Bob: 1, Carol: 0)
    (Alice: 0, Bob: 1, Carol: 1)

If the first combination was shown, the answers must have an even sum, otherwise the answers must have an odd sum.

First, without using probability, if A0 is the answer Alice gives when shown 0 and A1 when shown 1 (and similarly for Bob and Carol), you get the following set of equations needed to win always:

    A0 + B0 + C0 = 0      (mod 2)
    A1 + B0 + C1 = 1      (mod 2)
    A1 + B1 + C0 = 1      (mod 2)
    A0 + B1 + C1 = 1      (mod 2)

However, each term occurs twice on the left hand side, so when you add all equations up (mod 2) you would find the left hand side is even. However the right hand side sums to an odd number, thus all four equations can't hold simultaneously.

Now, probability doesn't help here, because any mixed strategy can be shown to be equivalent to a combination of pure strategies, none of which can guarantee a win.

Yet with a shared entangled state, it can be won 100% of the time. This isn't a probability thing - you can actually guarantee a win using the quantum strategy, even over arbitrarily large distances.

Got it, thank you. That makes sense.

Even knowing what goes on, I don't think it makes sense, so you're further than I am :)

Well I do have a degree in physics. That helps :)